Normalized defining polynomial
\( x^{22} - 8 x^{20} + 24 x^{18} - 28 x^{16} - 12 x^{14} + 66 x^{12} - 48 x^{10} - 28 x^{8} + 43 x^{6} + \cdots + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1288033148279983900699565800554496\) \(\medspace = -\,2^{22}\cdot 17524013040643^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(31.99\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(17524013040643\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $16$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $a^{21}-8a^{19}+24a^{17}-28a^{15}-12a^{13}+66a^{11}-48a^{9}-28a^{7}+43a^{5}-3a^{3}-7a$, $a^{20}-7a^{18}+17a^{16}-11a^{14}-23a^{12}+43a^{10}-5a^{8}-33a^{6}+10a^{4}+7a^{2}$, $a^{4}-a^{2}-1$, $2a^{21}-14a^{19}+35a^{17}-27a^{15}-39a^{13}+88a^{11}-23a^{9}-62a^{7}+33a^{5}+14a^{3}-6a$, $2a^{21}-14a^{19}+35a^{17}-26a^{15}-45a^{13}+101a^{11}-33a^{9}-69a^{7}+50a^{5}+9a^{3}-9a$, $a^{21}-7a^{19}+18a^{17}-17a^{15}-11a^{13}+38a^{11}-21a^{9}-11a^{7}+11a^{5}-4a^{3}+2a$, $a^{21}-7a^{19}+17a^{17}-10a^{15}-29a^{13}+56a^{11}-15a^{9}-40a^{7}+26a^{5}+4a^{3}-3a$, $a^{20}-6a^{18}+11a^{16}+2a^{14}-33a^{12}+35a^{10}+15a^{8}-41a^{6}+7a^{4}+11a^{2}-1$, $a^{21}-6a^{19}+11a^{17}+2a^{15}-34a^{13}+40a^{11}+7a^{9}-39a^{7}+16a^{5}+4a^{3}-4a$, $a+1$, $2a^{21}-2a^{20}-12a^{19}+12a^{18}+23a^{17}-23a^{16}-3a^{15}+3a^{14}-48a^{13}+48a^{12}+53a^{11}-53a^{10}+20a^{9}-20a^{8}-49a^{7}+49a^{6}+a^{5}-a^{4}+10a^{3}-10a^{2}+a$, $a^{21}-a^{20}-7a^{19}+6a^{18}+17a^{17}-11a^{16}-11a^{15}-2a^{14}-23a^{13}+33a^{12}+43a^{11}-35a^{10}-5a^{9}-15a^{8}-33a^{7}+41a^{6}+10a^{5}-7a^{4}+7a^{3}-11a^{2}+a+1$, $2a^{21}+a^{20}-13a^{19}-5a^{18}+28a^{17}+6a^{16}-9a^{15}+7a^{14}-56a^{13}-20a^{12}+78a^{11}+2a^{10}+10a^{9}+27a^{8}-74a^{7}-7a^{6}+17a^{5}-17a^{4}+19a^{3}-a+1$, $3a^{21}+a^{20}-21a^{19}-6a^{18}+53a^{17}+12a^{16}-44a^{15}-4a^{14}-50a^{13}-21a^{12}+126a^{11}+30a^{10}-44a^{9}-a^{8}-73a^{7}-20a^{6}+44a^{5}+10a^{4}+11a^{3}+a^{2}-6a-2$, $a^{21}+a^{20}-9a^{19}-7a^{18}+29a^{17}+18a^{16}-34a^{15}-18a^{14}-20a^{13}-5a^{12}+91a^{11}+25a^{10}-58a^{9}-11a^{8}-53a^{7}-4a^{6}+59a^{5}-5a^{4}+6a^{3}+a^{2}-10a+3$, $5a^{21}-a^{20}-36a^{19}+8a^{18}+93a^{17}-23a^{16}-76a^{15}+23a^{14}-103a^{13}+19a^{12}+252a^{11}-63a^{10}-86a^{9}+30a^{8}-171a^{7}+40a^{6}+115a^{5}-32a^{4}+29a^{3}-6a^{2}-23a+8$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 309215215.365 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{5}\cdot 309215215.365 \cdot 1}{2\cdot\sqrt{1288033148279983900699565800554496}}\cr\approx \mathstrut & 0.172793193006 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
A non-solvable group of order 81749606400 |
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ |
Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
11.9.17524013040643.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Degree 44 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | $16{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $22$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | $22$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.10.9 | $x^{10} + 10 x^{9} + 22 x^{8} - 8 x^{7} + 72 x^{6} + 1328 x^{5} + 4496 x^{4} + 3680 x^{3} - 3696 x^{2} - 96 x - 32$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
2.12.12.20 | $x^{12} - 2 x^{11} - 172 x^{9} - 172 x^{8} + 2112 x^{7} + 7968 x^{6} + 13568 x^{5} + 22320 x^{4} + 29216 x^{3} + 22912 x^{2} + 18624 x + 11200$ | $2$ | $6$ | $12$ | 12T87 | $[2, 2, 2, 2, 2]^{6}$ | |
\(17524013040643\) | $\Q_{17524013040643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17524013040643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{17524013040643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{17524013040643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |